Wojciech Dybalski (Göttingen):

Harmonic analysis of local operators in massless free field theory

Abstract:

The decomposition of the spectral measure of a Hamiltonian into its pure point, absolutely continuous and singular continuous part was crucial for the formulation and resolution of the problem of asymptotic completeness in quantum mechanics. Therefore, a deeper understanding of particle aspects of quantum field theory requires the development of an analogous, detailed theory of the spectrum of automorphism groups (i.e. the energy-momentum transfer of observables). This talk, which focuses on the momentum transfer of local observables in massless free field theory, reports on some work in this direction: It is shown that Wick powers and derivatives of the field have more regular behavior at small momentum transfer than the field itself. Exploiting this fact and phase space properties of the theory, subspaces are constructed in the local algebra of observables which contain operators of increasingly mild infrared behavior. These 'spectral subspaces' can be described with the help of a distinguished family of normal functionals which plays a role of the spectral measure. As an application we exhibit two sub-algebras, (invariant under certain gauge transformations), which have better spectral properties than the full local algebra.

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